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Question
Write the domain of the real function
`f (x) = 1/(sqrt([x] - x)`.
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Solution
\[Case-1: Whenx > 0\]
\[\left| x \right| = x\]
\[ \Rightarrow \frac{1}{\sqrt{\left| x \right| - x}} = \frac{1}{\sqrt{x - x}} = \frac{1}{0} = \infty \]
\[{Case-2: \text{Whenx} < 0\]
\[\left| x \right| = - x\]
\[ \Rightarrow \frac{1}{\sqrt{\left| x \right| - x}} = \frac{1}{\sqrt{- x - x}} = \frac{1}{\sqrt{- 2x}} \left( \text{exists because when} x<0, -2x>0 \right)\]
\[\ \text{Rightarrowf}\left ( x \right) \text {is defined whenx}< 0\]
\[\text{So, domain}=\left( - \infty , 0 \right)\]
\[\]
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